Can a "Blender Game Engine" powered game run on Linux?
Yes. Blender Game Engine powered games can run on all major platforms, and support for Android deployment, although limited and regarded as experimental by the community at large, is gaining ground. Deployment to Linux based systems is fully supported, and is a primary focus of the Blender Game Engine.

Keep in mind that a game created with the Blender Game Engine must be built for the system in question by someone with rightful access to the source files. This means that a developer must build their game for Linux in order for the game to run in a Linux environment. The same holds true for the other available platforms. A game built for Windows will only run in a Windows environment. 

One exception is Android, where (correct me if I am wrong) support only exists through a native application which requires the source file in .blend format. Again, this functionality is regarded as experimental and as such should not be considered a viable deployment solution at this time. The Blender Foundation provides limited insight on this subject, however relevant information can be found in the Official Documentation here and here. 

From the Official Documentation for the Blender Game Engine:


  The Game Engine can simulate content within Blender,   however it also
  includes the ability to export a binary   run-time to Windows, Linux
  and MacOS. There is also basic   support for mobile platforms with the
  Android Blender   Player GSOC 2012 project.